Mission Impossible – or – You Do the Math

I have made this [set of articles] longer than usual, because I lack the time to make it short.
Pascal

Barry* and Kelly gave me a math book, Fearless Symmetry (Exposing the Hidden Patterns of Numbers) [commercial site]. Fearless proposes to introduce to non-specialists the basic concepts of algebraic number theory (ANT), Barry’s field.

I plan to write some notes to myself during my study of Fearless, motivated by my own peculiar need to talk myself through the subject, and to bring necessary notions into the mix that are not made explicit in Fearless. These notes will be subjects of constant ‘improvement’ as my knowledge (hopefully) increases. Welcome to the world’s longest book report ever, written sequentially in a number of installments. This is the first of the series, and deals with preparation, goals, and some human interest background.

Fearless is well-written, but absent prior knowledge of mathematical symbolism and the fundamentals of abstract number systems, the book will become tough sledding in its more advanced treatments. The Fearless authors state that the reader should have mathematical sophistication roughly equivalent to having successfully completed a course in calculus.

An attempt is made to define needed concepts within the book, but it seems some terms are referenced before the definition appears, and the index is spotty. A reader without prerequisite mathematical exposure will find it difficult to acquire all this conceptual machinery for the first time and then to use it while following the development.

My mission (which I have accepted) is to understand a bit about the appeal of ANT. However, there is no expectation, nor even hope, that I will understand anything about Barry’s specialized investigations in ANT. I can read his thesis with comprehension only up to about page 4 (it’s 180 pages long). After I get through Fearless, I will still be on page 4. But I should have a feel for the subject that is missing now.

The Fearless content is in three parts:

  1. Algebraic Preliminaries
  2. Galois Theory and Representations
  3. Reciprocity Laws

The first part of Fearless was pretty well covered in my university-level mathematics. By the end of the second part, true comprehension will be beyond the reach of most without a solid background in abstract mathematics. But short of comprehension, most may be able to come away with a sense of the lay of the land through which the mathematicians passed to reach the proof of Fermat’s last conjecture.

The only new concepts for me in the first two parts are ‘variety’ and ‘representation’.  I will try to use Fearless’ introduction to these topics to prepare for entering  into deeper expositions in books in my library. I will write more about my Fearless journey when I finish a few chapters with measurable comprehension. BTW, I always accept donations of math books.

* Human interest digression:

In mathematical genealogy, one means by father:son the relation of PhD-advisor:PhD-candidate. Barry’s mathematical genealogy is Popescu-Rubin-Wiles-Coates-Baker-Davenport-Littlewood. This is a highly distinguished lineage (Fields Medal, two Cole Prizes, a world-renowned proof of 400 yo conjecture, and very numerous other awards and prestigious offices) that goes back about 100 years in England.

The Rouse Ball Chair in Mathematics at Cambridge was held by Littlewood and Davenport for over thirty years (not consecutively, Besicovitch intervened). John Littlewood was noted for his work in dynamical systems. He bequeathed us the Littlewood inequality and the Hardy-Littlewood conjectures.

Harold Davenport worked in analytic number theory; his text, Multiplicative Number Theory, is a classic. He also wrote a classic (and still available) expository number theory book for general audiences, The Higher Arithmetic.

Alan Baker, Emeritus Professor of Pure Mathematics at Cambridge, is noted for work in transcendence theory and Diophantine equations. He conjectured and then proved Baker’s Theorem, providing generalized results related to Hilbert’s 7th problem, for which he was awarded the Fields Medal. He further compiled complete lists for class numbers 1 and 2, related to Gauss’s class number problem. His text, Transcendental Number Theory, introduces these results, among others.

Beginning in the 1960s, a research track in algebraic number theory concerned itself with L-functions, leading to conjectures such as those of Birch and Swinnerton-Dyer, and of Brumer-Stark. Iwasawa Theory developed as an abstract tool for attacking such problems. The lineage of mathematical descendants of Alan Baker focused on such problems, aiming ultimately for the great open conjectures, Fermat’s Last Conjecture and the Riemann Hypothesis.

John Coates was Sadleirian Professor of Pure Mathematics at Cambridge for 25 years (a chair previously held by Cayley and Hardy). He works in Iwasawa Theory and contributed to the Coates-Sinnot conjecture. Working with his student Wiles, they proved a partial case of the Birch and Swinnerton-Dyer conjecture. (On a more personal note, Professor Coates, a true gentleman, endeared himself to me by answering an off-the-wall email**.

Andrew Wiles went on to prove Fermat’s last conjecture as well as contributing to a proof of the main conjecture of Iwasawa Theory. His research prompted the book Fearless Symmetry. He is now a Professor at Oxford.

Karl Rubin was Barry’s post-doctoral advisor and colleague at UCI, where he is Thorp Professor of Mathematics. He researches elliptic curves and associated Euler/Kolyvagin systems, which has lead to a more elementary proof of the main conjecture of Iwasawa Theory.

Christian Popescu was Barry’s thesis advisor at UCSD and is an active researcher in p-adic L-functions, the Brumer-Stark conjecture, and Iwasawa theory. Barry follows in these learned research footsteps, working directly on the Brumer-Stark conjecture while also working to improve pedagogy in basic number theory by charting avenues suitable for undergraduate research.

In addition to his mathematical genealogy, Barry also has a distinguished lineage in his undergraduate major of chemical physics. His research advisor, Professor W. E. Moerner, now at Stanford, was a 2014 Chemistry Nobelist. Barry is listed as co-author with Prof. Moerner on several papers researching photorefractive polymers. This research is surprisingly germane to the current discussion because, through this non-mathematical collaboration, Barry has acquired a finite Erdős number, currently bounded at 7.

** Here’s a personal anecdote, the highlight of a stupendously silly investigation I conducted:

While proofreading Barry’s thesis (only the wordy part, let me be explicit), I encountered the usage ‘associated to’. I had learned ‘associated with’. Curiosity aroused, I checked a math publication by his advisor, C. Popescu, and saw that he had also embraced the ‘odd’ usage. So it seemed that perhaps Barry picked it up there, due to the author having a non-English primary language. But on the chance the curiosity might extend deeper, I found somewhere a paper by K. Rubin and there it was, ‘associated to’. ‘Curiouser and curiouser’, I next checked some writing by A. Wiles. Again, ‘associated to’. I was now assuming that all mathematicians were taught from a different primer, so I checked the math books in my library. Not a single instance of ‘associated to’ was in evidence. (Good editors are worth their weight in math books – the new gold standard)***.

Back on the original trail, I tried to find a paper by J. Coates on the Internet, but found none. So I did the next best thing and wrote to Professor Coates at Cambridge, asking him if he used ‘…to’ or ‘…with’. He VERY kindly wrote back that although his origins in Australia might appear to make him a suspect, he was taught ‘associated with’. So the chain was broken.

I am disappointed because I was imagining that there was a neurological basis for this usage in very bright minds. But apparently it is just a case of random rebellions at grammatical rules and I will have to leave it at that. But feel free to pick up this ball and run with it. (I still feel that I was on to something deep.) As for Barry’s thesis, it is letter perfect so far as I can tell; every ‘associated to’ has been mapped to normalized English. I hope Barry gets a chance some day to meet Professor Coates.

***Postscript: Well, guess what? One or more of the Fearless authors uses ‘associated to’, the first published book in which I have encountered the odd usage. I am shocked, SHOCKED, at such a lack of competent editing. (Of course, change is part of metaphysics, those exercises of the human mind, such as language, that follow various paths of meme evolution. But one would hope that editors would be our last bastion for preventing the clutter of too many non-poetic ways to express the same thought.)

BTW, I was perusing Barry’s current preprints and was very pleased that ‘associated with’ is winning out. One ‘to’ made it through, but that’s ok. Perfection is overrated, and Barry is converging.

YAP (yet another postscript): Barry forwarded to me a brief style guide for LaTex publishing. Check out 2.6. Shockingly, my pedantry has been formalized.

Proceed to Representations, Groups, Permutations.

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